Introduction

This project provides a native .NET implementation for various variants of binary decision diagrams. It focuses on high performance, usability, and correctness.

Installation

Just add the project to your visual studio solution or add the package from nuget.

Getting Started

To import the library, add the following line to your file:

using DecisionDiagrams;

A basic use of the library is shown shown below:

// create a manager that uses traditional binary decision diagrams.
// alternatively use `new DDManager<CBDDNode>()` for the chain-reduced variant.
var manager = new DDManager<BDDNode>();

// allocate three variables, two booleans and one 32-bit integer
// the internal ordering will match the order allocated from the manager.
var a = manager.CreateBool();
var b = manager.CreateBool();
var c = manager.CreateInt32();

// take the logical or of two variables.
// the Id() method returns the identity DD for the variable.
DD f1 = manager.Or(a.Id(), b.Id());

// create a DD representing whether 1 <= c <= 4 and c != 1.
// a bitvector represents a set of integers by using one DD per bit in the integer.
DD f2 = manager.GreaterOrEqual(c.ToBitvector(), manager.CreateBitvector(1));
DD f3 = manager.LessOrEqual(c.ToBitvector(), manager.CreateBitvector(4));
DD f4 = manager.And(f1, manager.And(f2, f3));

// get a satisfying assignment for a formula.
// will be null if no assignment exists.
var assignment = manager.Sat(f4);

// get the values back as C# objects
bool valuea = assignment.Get(a);  // valuea = false
bool valueb = assignment.Get(b);  // valueb = true
int valuec = assignment.Get(c);   // valuec = 1

API Examples

Logical Operations: The library supports negation, disjunction, conjunction, if-then-else, if-and-only-if, and implication.

var a = manager.CreateBool();
var b = manager.CreateBool();
var c = manager.CreateBool();

DD f1 = manager.And(a.Id(), b.Id());         // a and b
DD f2 = manager.Or(a.Id(), b.Id());          // a or b
DD f3 = manager.Not(a.Id());                 // not a
DD f4 = manager.Implies(a.Id(), b.Id());     // a implies b
DD f5 = manager.Iff(a.Id(), b.Id());         // a == b
DD f6 = manager.Ite(a.Id(), b.Id(), c.Id()); // If a then b else c.

Variable Quantification and Substitution: Existential & Universal quantification are supported, as well as variable substitution.

var a = manager.CreateBool();
var b = manager.CreateBool();
var c = manager.CreateBool();

// a and b and c
DD f1 = manager.And(a.Id(), manager.And(b.Id(), c.Id()));
// existential quantification. result = c
DD f2 = manager.Exists(f1, manager.CreateVariableSet(a, b));
// replace a with c. result = b and c
DD f3 = manager.Replace(f1, manager.CreateVariableMap(new Dictionary<Variable<BDDNode>, Variable<BDDNode>> { { a, c } }));

Arithmetic Operations: The library supports integer arithmetic operations through a Bitvector abstraction, which uses one DD per bit in the integer. A Bitvector represents a set of integer values.

// create a single boolean variable.
var g = manager.CreateBool();

// create two 16-bit integer variables with an interleaved variable ordering.
// an interleaved ordering tends to perform the best for integer variables that are
// combined with other variables using arithmetic or comparision operations.
var intvars = manager.CreateInterleavedInt16(2);
var a = intvars[0];
var b = intvars[1];

BitVector<BDDNode> bv1 = manager.And(a.ToBitvector(), b.ToBitvector());         // bitwise and (&)
BitVector<BDDNode> bv2 = manager.Or(a.ToBitvector(), b.ToBitvector());          // bitwise or (|)
BitVector<BDDNode> bv3 = manager.Not(a.ToBitvector());                          // bitwise negation (~)
BitVector<BDDNode> bv4 = manager.Add(a.ToBitvector(), b.ToBitvector());         // addition (+)
BitVector<BDDNode> bv5 = manager.Subtract(a.ToBitvector(), b.ToBitvector());    // subtraction (-)
BitVector<BDDNode> bv6 = manager.Ite(g.Id(), a.ToBitvector(), b.ToBitvector()); // if g then a else b

DD bv7 = manager.Eq(a.ToBitvector(), b.ToBitvector());                          // equality (=)
DD bv8 = manager.Less(a.ToBitvector(), b.ToBitvector());                        // less than (<)
DD bv9 = manager.LessOrEqual(a.ToBitvector(), b.ToBitvector());                 // less than or eq (<=)
DD bv10 = manager.LessOrEqualSigned(a.ToBitvector(), b.ToBitvector());          // less than or eq signed (<=)
DD bv11 = manager.Greater(a.ToBitvector(), b.ToBitvector());                    // greater than (>)
DD bv12 = manager.GreaterOrEqual(a.ToBitvector(), b.ToBitvector());             // greater than or eq (>=)
DD bv13 = manager.GreaterOrEqualSigned(a.ToBitvector(), b.ToBitvector());       // greater than or eq signed (>=)

Variable Ordering and Interleaving: You can control the decision diagram variable ordering to improve performance in several ways:

// create a manager that uses traditional binary decision diagrams.
var manager = new DDManager<BDDNode>();

// the order will be 'a' before 'b'.
var a = manager.CreateBool();
var b = manager.CreateBool();

// all bit variables for 'c' will be after 'a' and 'b'.
// the variables in 'c' will be allocated in order of most significant bits first.
var c = manager.CreateInt64(BitOrder.MSB_FIRST);

// all bit variables for 'd' will be after all variables for 'c'.
// the variables in 'd' will be in order of least significat bits first.
var d = manager.CreateInt64(BitOrder.LSB_FIRST);

// the ith variable in 'e' will be in the (i + 3) % 64 position.
var e = manager.CreateInt64((i) => (i + 3) % 64);

// there will be two integers in 'f', each is a 64 bit integer.
// the bit ordering will alternate between the two variables with the MSB ordering.
// x0, y0, x1, y1, ..., x63, y63
var f = manager.CreateInterleavedInt64(2, BitOrder.MSB_FIRST);
var f0 = f[0];
var f1 = f[1];

Implementation Details

The library is based on the cache-optimized implementation of decision diagrams here, and implements two variants:

• Binary decision diagrams (link)
• Chain-reduced binary decision diagrams (link)

Data representation

Internally decision diagram nodes are represented using integer ids that are bit-packed with other metadata such as a garbage collection mark bit, and a complemented bit. User references to nodes (DD type) are maintained through a separate (smaller) table.

Garbage collection

The DD reference table uses WeakReference wrappers to integrate with the .NET garbage collector. This means that users of the library do not need to perform any reference counting, which is common in BDD libraries. Nodes are kept in a memory pool and the library maintains the invariant that a node allocated before another will appear earlier in this pool. This allows for various optimizations when looking up nodes in the unique table. To uphold this invariant, the library implements a mark, sweep, and shift garbage collector that compacts nodes when necessary.

Memory allocation

By hashconsing nodes in the unique table, the library ensures that two boolean functions are equal if and only if their pointers (indices) are equal. The unique table holds all nodes and is periodically resized when out of memory. For performance reasons, we ensure that this table is always a power of two size. This makes allocating new space a bit inflexible (harder to use all memory) but in return makes all operations faster. To compensate for this inflexible allocation scheme, the library becomes more reluctant to resize the table as the number of nodes grows.

Optimizations

The library makes use of "complement edges" (a single bit packed into the node id), which determines whether the formula represented by the node is negated. This ensures that all negation operations take constant time and also reduces memory consumption since a formula and its negation share the same representation. The implementation also includes a compressed node type CBDDNode that should offer lower memory use and often higher performance but comes with the restriction that you can not create more than 2^15-1 binary variables.

Operations

Internally, the manager supports several operations: conjunction, existential quantification, if-then-else and then leverages free negation to support other operations efficiently. It also leverages commutativity of conjunction + disjunction to further reduce memory by ordering the arguments to avoid redundant entries. Currently, the library does not support dynamic variable reordering as well as a number of operations such as functional composition.

Performance

The performance of the library should be comparable to other highly optimized BDD implementations. Below are the timings to solve the famous n-queens chess problem (how to arrange n queens on an n x n chess board such that none attack each other). The library is compared to BuDDy, which is considered to be one of the fastest BDD implementations, as well as JavaBDD, which has a direct translation of the C-based BuDDy implementation into Java. The times given are using .net core 6.0 for a 64-bit Intel Core i7-8650U CPU @ 1.90GHz machine. All implementations require around 200MB of memory, while the CBDDNode implementation uses roughly half that at 100MB of memory.

Implementation	Language	n	Time (seconds)
DecisionDiagrams (CBDDNode)	C#	12	11.4s
DecisionDiagrams (BDDNode)	C#	12	14.8s
BuDDy (aggressive allocation)	C	12	21.9s
JavaBDD (BuDDy translation)	Java	12	27.5s
BuDDy (default settings)	C	12	35.9s

Contributing

This project welcomes contributions and suggestions. Most contributions require you to agree to a Contributor License Agreement (CLA) declaring that you have the right to, and actually do, grant us the rights to use your contribution. For details, visit https://cla.opensource.microsoft.com.

When you submit a pull request, a CLA bot will automatically determine whether you need to provide a CLA and decorate the PR appropriately (e.g., status check, comment). Simply follow the instructions provided by the bot. You will only need to do this once across all repos using our CLA.

This project has adopted the Microsoft Open Source Code of Conduct. For more information see the Code of Conduct FAQ or contact opencode@microsoft.com with any additional questions or comments.